Simple explicit method

How does one discretise a 2-D system Assume the diffusion equation 

The simplest measure for conserving computer time would be to introduce unequal intervals. For a disk, for example, the Feldberg (1981) transformation Eq. 5.69 could be used in the z-direction (see Fig. 8.1). In the r-direction, points could be widely spaced near the origin but closely spaced on both sides of the disk edge (radius A compression function in r around rQ, similar to Eq. 5.69, is 

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Most of the practical electrode geometries (microdisc, microband, channel, wall jet) require simulation of two spatial dimensions. Although a few early simulations used a simple explicit method (Britz, 1988), its relative inefficiency is compounded in multiple dimensions. Two ways of adding some implicit character to multidimensional simulations have been adopted ... 

Finite difference methods have been used bpth to test the assumptions made in the derivation of eqn. (27) under the Leveque approximation [35] and to solve electrochemical diffusion-kinetic problems with the full parabolic profile [36-38]. The suitability of the various finite difference methods commonly encountered has been thoroughly investigated by Anderson and Moldoveanu [37], who concluded that the backward implicit (BI) method is to be preferred to either the simple explicit method [39] or the Crank-Nichol-son implicit method [40]. 

Exercise 8.1 Rework the problem of Example 8.1 using the simple explicit method and the following initial and boundary conditions ... 

The two ways of learning - deductive and inductive - have already been mentioned. Quite a few properties of chemical compounds can be calculated explicitly. Foremost of these are quantum mechanical methods. However, molecular mechanics methods and even simple empirical methods can often achieve quite high accuracy in the calculation of properties. These deductive methods are discussed in Chapter 7. 

A seemingly simple measurement method to quantify the burning rate (conversion gas rate) of a simulated crosscurrent moving bed was obtained by Lamb et al. However, the mathematical relationship between burning rate and measurands was not explicitly declared. No verification method is used and no uncertainty analysis is carried out. The method is badly defined. Consequently, the results would be difficult to reproduce. 

In principle, all the methods described above for single odes can be used for the solution of such a system, when extended suitably. In the case of explicit methods such as Euler or RK, this is very simple to implement, whereas with implicit methods such as BI or the trapezium method, there are some choices to be made. 

One of the main uses of digital simulation - for some workers, the only application - is for linear sweep (LSV) or cyclic voltammetry (CV). This is more demanding than simulation of step methods, for which the simulation usually spans one observation time unit, whereas in LSV or CV, the characteristic time r used to normalise time with is the time taken to sweep through one dimensionless potential unit (see Sect. 2.4.3) and typically, a sweep traverses around 24 of these units and a cyclic voltammogram twice that many. Thus, the explicit method is not very suitable, requiring rather many steps per unit, but will serve as a simple introduction. Also, the groundwork for the handling of boundary conditions for multispecies simulations is laid here. 

This is a simple simulation of a CV experiment, using the explicit method EX, and assuming a quasireversible reaction,... 

These relationships link variations in the shape function to the reactivity descriptors commonly used in DFT and provide explicit methods for describing chemical reactivity in terms of the shape function yielding conceptual shape function theory . Thus, it is clearly seen that the shape function not only determines all the physical properties of an isolated molecule but, since reactivity descriptors can be explicitly constructed from the shape function, also its chemical properties. As the resulting equations are not always that simple to apply at first sight, we pass in next paragraph to some pragmatic procedures for extracting descriptors from the shape function. 

The simple, explicit difference method for transient heat conduction problems... 

A very thick wall with constant thermal diffusivity a and a constant initial temperature do is heated at its surface. The temperature rises there, between t = 0 and t = t linearly with time t, to the value di > do, which remains constant for t > t. The temperature profile in the wall at times t = f and t = 2f is to be calculated numerically. The simple explicit difference method is to be used, with At = t /6 and M = 1/3, and the normalised temperature d+ = (d — do)/(di — do) is to be used. Compare the numerically calculated values with the exact solution... 

In Eq. (2.110), we deliberately ignored additional terms arising from the permutation of triplets of momenta, quadruplets, and so on because we will be interested mostly in situations in which 2> A. Because these higher-order terms involve sums over products of three and more factors of the form exp (—Trr /A ), their contribution to the semiclassic correction to Qci vanishes rapidly. We note in passing that a simple graphical method can be devised to derive explicit forms for the contributions from triplet, quadruplet, and so on permutations. A detailed discussion of this technique is, however, beyond the scope of this chapter. 

The above illustrates that the explicit method provides a simple and readily implemented approach to simulate the course of reaction kinetics, even for rather complicated reaction mechanisms, as long as the initial concentrations of the reactants and products, and the rate constants, are known. The problem need not have a known, closed-form mathematical solution. Non-linear relations are no impediment to the simulation, because the equations are linearized. Such linearization is an acceptable approximation as long as the time increments At are sufficiently small. In section 9.2d we have seen how we can exploit a user-defined function to make the increments At smaller, by moving some of the computations off the spreadsheet, without having to change either the time range covered or the column length used. In principle, the same method can of course be... 

The first application of quantitative qnantnm theory to chemical species significantly more complex than the hydrogen atom was the work of HiickeP on unsaturated organic componnds, in 1930-1937 [19], This approach, in its simplest form, focuses on the p electrons of double bonds, aromatic rings and heteroatoms. Althongh Hiickel did not initially explicitly consider orbital hybridization (the concept is nsnally credited to Panling, 1931 [20]), the method as it became widely applied [21] confines itself to planar arrays of -hybridized atoms, nsnally carbon atoms, and evaluates the consequences of the interactions among the p electrons (Fig. 4.4). Actually, the simple Hiickel method has been occasionally applied to nonplanar systems [22]. Because of the importance of the concept of hybridization in the simple Hiickel method a brief discussion of this concept is warranted. 

The most sophisticated treatment of this problem to date is that due to Aoki et al. [62], at the Levich approximation level, for reversible electron transfer. Figure 17 shows the resulting dimensionless chronoam-peromogram, in terms of the normalised time, t. The curve is identical to that obtained by Flanagan and Marcoux using the simple explicit numerical method [35]. At short times (t < 0.24), the behaviour is Cottrellian and the equation... 

In simpler stationary cases the author would always prefer solutions obtained by applying the Press-P9-method (compare Section 15.3.1), simple explicit numerical two-step models (compare Section 15.3.2), and numerical solutions, particularly for non-stationary cases or in cases with complex reactions, by using models like STEADYSEDl or CoTReM (compare Section... 

The oscillator model for proton transfer was first developed by DOGONADZE and KUSNETSO /147/. The above treatment proposed by CHRISTOV /37e/ is based on the general "theory of reaction rates applied to the two-frequency oscillator model. It reproduces the essential results of earlier work concerning electronically and protonically non-adiabatic reactions and yields, moreover, simple, explicit expressions for adiabatic reactions never derived before. This shows the utility of certain new methods in calculating reaction probabilities, developed in Chapter II, which allow an application of the most suitable formulations of the rate theory. 

This is a simple equation to solve, and each value of can be solved for independently at every timestep because all of the terms appearing on the right-hand side of the equation are fully known (this is in contrast with the implicit method which we shall come to shortly). Simulation using the explicit method simply consists of iterating from timestep 0, where for all... 

There are some fundamental approximations in the simple LCAO method that are harder to evaluate. One is the validity of the linear combination of atomic orbitals as an approximation to molecular orbitals. Another is the assumption of localized o bonds. A proper treatment probably should take account of the so-called cr—ir interactions. Beyond these rather basic assumptions is the bothersome business of dealing explicitly with interelectronic repulsions. These repulsions are expected to be functions of molecular geometry as well as the degree of self-consistency of the molecular field. Thus, cyclobutadiene must have considerably greater interelectronic repulsion than butadiene, with the same number of tr electrons. 

Fletcher (1974) introduced unequal 8x intervals Whiting and Carr (1977) applied orthogonal collocation to electrochemistry Shoup and Szabo (1982) applied Gourlay s (1970) hopscotch method to electrochemistry and Heinze et al (1984) showed how to include the boundary value c in the implicit equations of the Crank-Nicolson method, thereby removing a major problem with that method. Britz (1988) applied simple explicit... 

We assume, for a start, the simple diffusion equation 5.12. We have seen in Sect. 5.1, that the normal explicit method, with its forward-difference discretisation of 8c/3t performs rather poorly, with an error of 0(6t). The discrete expression for the second derivative (right-hand side of Eq. 5.12) is better, with its error of 0(h ). Let us now imagine a time t+ig6t at this time, the discretisation... 

Hiickel found that, by treating only the n electrons explicitly, it is possible to reproduce theoretically many of the observed properties of unsaturated molecules such as the uniform C-C bond lengths of benzene, the high-energy barrier to internal rotation about double bonds, and the unusual chemical stability of benzene. Subsequent work by a large number of investigators has revealed many other useful correlations between experiment and this simple HMO method for n electrons. 

Following Huckel, we ignore all the a-type AOs and take the three 2pz AOs as our set of basis functions. Notice that this restricts us to the carbon atoms the hydrogens are not treated explicitly in the simple HMO method. We label our three basis functions Xu X2, X3 as indicated in Fig. 8-2. We will assume these AOs to be normalized. 

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